Kyle Barker, PhD
Assistant Professor & Graduate Coordinator
University of Arkansas at Little Rock
Email: wkbarker@ualr.edu
Research
My primary research focus is on the existence and dynamics of traveling wave solutions in reaction-diffusion equations with delay. In particular, I am especially interested in systems where the delay appears in the diffusion term. Topics I look forward to exploring include the stability of such systems, the spreading speed of traveling waves in biological and chemical models, and the existence of traveling wave solutions in equations with non-constant diffusion coefficients. My current approach draws on techniques from functional and complex analysis, particularly fixed-point iteration and the method of upper and lower solutions, as well as finite difference numerical methods. I am also interested in the broader dynamics of evolution equations. Recently, I have been studying traveling wave dynamics in models with cross-diffusivities, as well as nonclassical diffusion models where the time evolution of spatial curvature plays a role in the system's behavior.
In our most recent work, we studied the existence and uniqueness of bounded asymptotic mild solutions to evolution equations of the form u'(t) = Au(t) + f(t) in Banach spaces, where \(A \) generates an (analytic) \( C_0 \)-semigroup and \( f \) is bounded. We established spectral conditions on \( A \) and \( f \) ensuring solutions with the same asymptotic profile as \( f \), including resonance cases via Massera-type theorems. These results extend classical theories and have applications to PDEs.
Papers and Preprints:
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W.K. Barker, V.T. Long, D.H. Nguyen, M.V. Nguyen, Existence of Bounded Asymptotic Solutions of Autonomous Differential Equations, Journal of Differential Equations, (434) (2025) 113320.
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W.K. Barker and M.V Nguyen, Traveling Waves in Reaction-Diffusion Equations with Delay in the Diffusion Term , Journal of Nonlinear Analysis 271 (2026) 114125. https://doi.org/10.1016/j.na.2026.114125
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W.K. Barker, Existence of Traveling Waves of Lokta Volterra Type Models with Delayed Diffusion Term and Partial Quasimonotonicity, International Journal of Applied and Computational Mathematics, (12) 43 (2026). https://doi.org/10.1007/s40819-026-02113-x
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W.K.Barker and A. Embry, Traveling Wave Solutions for a Nonlocal Dispersal SIR Model with Delayed Transmission, AIMS Mathematics, 11(5) 12795-12824 (2026) 10.3934/math.2026527
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W.K. Barker, Existence of Traveling Waves in a Nicholson Blowflies Model with Delayed Diffusion Term, Proceedings of the American Mathematical Society, (2025). https://doi.org/10.1090/proc/17233
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W.K. Barker, M. V. Nguyen , and N. T. Thanh, Traveling waves for nonlocal Fisher-KPP equations with Diffusive Delay, Journal of Evolution Equations, 25 (38) (2025). https://doi.org/10.1007/s00028-025-01064-y
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W.K. Barker. Existence of Traveling Waves in a SIR Model with incidence and Delayed Diffusion Term, Mathematics and Computers in Simulation, (2025) https://doi.org/10.1016/j.matcom.2025.04.027.
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W.K. Barker, M. V. Nguyen , and N. T. Thanh, On Monotone Traveling Waves of Mackey Glass Equation with Delay in Diffusion, Journal of Applied Analysis, DOI: 10.1515/jaa-2024-0188 (2025).
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W.K Barker, T.D Nguyen, D.T Nguyen, K. Seo. Rigidity Properties of P-Biharmonic Maps and P-Biharmonic Submanifolds, Journal of Mathematical Analysis and Applications, 537 (2), (2024) 128310.
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W.K.Barker and A. Simms, Existence of Traveling Waves in a Predator-Prey Invasion Model with Nonlocal Dispersal and Delayed Effects in Dispersal, accepted, Open Journal of Mathematical Analysis (2025).
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A. Alzahrani, W.K Barker. Monotone Traveling Waves in a General Discrete Integral Difference Model for Populations, Malaya Journal of Matematik, 11 No. S (2023) .
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W.K. Barker and N. T. Thanh, On Monotone Traveling Waves of Bud Worm Spruce Equation with Delay in Diffusion, submitted.
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W.K. Barker, L.X. Dong, V. T. Luong, N.D. Toan, Traveling waves in Nonclassical Diffusion Equations, submitted.
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W.K. Barker and A. Childers, Traveling Waves with nonlocal dispersal and response and delay, submitted.
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W.K. Barker, X. Ye, S. Zhang, Weak Galerkin FEM for Delayed Reaction–Diffusion Equations, submitted.
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W.K. Barker, L.X. Dong, V. T. Luong, N.D. Toan, Traveling waves in Nonclassical Diffusion
Equations with delay, submitted. -
W.K. Barker and A. Sanders, Monotone Traveling waves in a general discrete model for populations
with k-generation long-term memory, submitted. -
W.K. Barker, Persistence and Stability of Solitary Waves in a Spatially Balanced Regularized
Long Wave Equation, submitted. -
W.K.Barker and A. Embry, Existence of Traveling Waves in a SIR Model with Nonlocal Dispersal and Delayed Effects in Dispersal, submitted.
Peer-Reviewed Journals Refereed For:
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Nonlinearity
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Journal of Evolution Equations
3. Mathematics and Computers in Simulation
4. Journal of Physics A: Mathematical and Theoretical
5. Advances in Nonlinear Analysis
6. Differential Equations and Dynamical Systems
7. Electronic Journal of Differential Equations
Students Advised:
Master Level Students
1.) Skye Fisher (2024): Strengthening Math Retention in College: Effective Strategies
for Long-Term Success. Instructor: University of Arkansas – Pulaski Technical College
2.) Austin Simms (2025): Existence of Traveling Waves in a Predator-Prey Invasion Model with Nonlocal Dispersal and Delayed Effects in Dispersal. Intructor: Henderson State University
3.) Greg Simpson (2025): On the Efficacy of Structured Mathematics and STEM Tutoring Centers: A Meta Analysis Study. PhD Student, University of Memphis
4.) Austin Sanders (2026): Monotone Traveling waves in a general discrete model for populations with k-generation long-term memory
PhD Level Students
1.) Ashley Embry, Traveling Wave Fronts for SIR Type Diffusion Models with Delayed Nonlocal Terms. (Spring 2026)
2.) Travis Chism, Symmetric Deformations of the Varga Material in Nonlinear Elasticity (Proposal Passed, Spring 2026)